When closed graph manifolds are finitely covered by surface bundles over \(S^1\)
type: Item
instance of: scholarly article
title: When closed graph manifolds are finitely covered by surface bundles over \(S^1\)
zbMATH Open document ID: 0919.57005
author: Yan Wang
author: Fengchun Yu
publication date: 1999-08-25T00:00:00Z
review text: A graph manifold is a 3-manifold constructed by identifying boundary components of compact Seifert-fibered 3-manifolds. From the authors' introduction: ``A classical fact is that Seifert manifolds with nonempty boundaries are finitely covered by surface bundles over \(S^1\). And a closed Seifert manifold is covered by a surface bundle over \(S^1\) if and only if its Euler number is zero. In the past several years, researchers have proven that some closed graph manifolds are not covered by surface bundles over \(S^1\) [\textit{J. Luecke} and \textit{Y. Q. Wu}, AMS/IP Stud. Adv. Math. 2 (pt. 1), 80-103 (1997; Zbl 0889.57024); \textit{W. D. Neumann}, Topology 36, No. 2, 355-378 (1997; Zbl 0872.57021)], and that graph manifolds with nonempty boundary are finitely covered by surface bundles over \(S^1\) [\textit{S. C. Wang} and \textit{F. C. Yu}, Math. Proc. Camb. Philos. Soc. 122, No. 3, 447-455 (1997; Zbl 0899.57011)]. In this paper we discuss the case when a closed graph manifold is finitely covered by a surface bundle over \(S^1\). For a given closed graph manifold \(M\), there are two associated matrices \(B(M)\) and \(W(M)\), where \(B(M)\) is defined in [\textit{S. C. Wang} and \textit{Y. Q. Wu}, Proc. Lond. Math. Soc., III. Ser. 68, No. 1, 203-224 (1994; Zbl 0841.57028)] and \(W(M)\) in [\textit{S. C. Wang} and \textit{F. C. Yu}, loc. cit. ]. We have two results. The first is Theorem 2.1, which is a necessary condition for \(M\) to be covered by a surface bundle. This condition can be read directly from the entries of \(B(M)\) and \(W(M)\), and is therefore very useful in practice. The second result is Theorem 3.1, which is a necessary and sufficient condition for \(M\) to be covered by a surface bundle. The condition is that a system of equations, whose coefficients are related to \(B(M)\) and \(W(M)\) in a complicated way, has a certain solution.'' This is not easy to compute in practice, in particular, it is difficult to show that the system does not have a re…
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 1279333
zbMATH Keywords: graph manifold
zbMATH Keywords: covering
zbMATH Keywords: surface bundle
reviewed by: Darryl McCullough
MaRDI profile type: Publication
full work available at URL: s10114-999-0057-5
OpenAlex ID: W2035106335
cites work: Q4382044
cites work: Commensurability and virtual fibration for graph manifolds
cites work: Graph manifolds with non-empty boundary are covered by surface bundles
cites work: Q4114528
cites work: The Geometries of 3-Manifolds
cites work: \(\pi_1\)-injective surfaces in graph manifolds
DOI: 10.1007/S10114-999-0057-5
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1279333